Theorem bnj154 30948

Description: Technical lemma for bnj153 30950. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj154.1	⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
bnj154.2	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Assertion

Ref	Expression
bnj154	⊢ (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑓,𝑔   𝑥,𝑓

Allowed substitution hints:   𝐴(𝑥,𝑔)   𝑅(𝑥,𝑔)   𝜑′(𝑥,𝑓,𝑔)   𝜑₁(𝑥,𝑓,𝑔)

Proof of Theorem bnj154

Step	Hyp	Ref	Expression
1		bnj154.1	. 2 ⊢ (𝜑1 ↔ [𝑔 / 𝑓]𝜑′)
2		bnj154.2	. . 3 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
3	2	sbcbii 3491	. 2 ⊢ ([𝑔 / 𝑓]𝜑′ ↔ [𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
4		vex 3203	. . 3 ⊢ 𝑔 ∈ V
5		fveq1 6190	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅))
6	5	eqeq1d 2624	. . 3 ⊢ (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)))
7	4, 6	sbcie 3470	. 2 ⊢ ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
8	1, 3, 7	3bitri 286	1 ⊢ (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))

Syntax hints:   ↔ wb 196   = wceq 1483  [wsbc 3435  ∅c0 3915  ‘cfv 5888   predc-bnj14 30754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj153  30950  bnj580  30983  bnj607  30986