Theorem bnj610 30817

Description: Pass from equality (𝑥 = 𝐴) to substitution ([𝐴 / 𝑥]) without the distinct variable restriction ($d 𝐴 𝑥). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj610.1	⊢ 𝐴 ∈ V
bnj610.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
bnj610.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓′))
bnj610.4	⊢ (𝑦 = 𝐴 → (𝜓′ ↔ 𝜓))

Assertion

Ref	Expression
bnj610	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)

Distinct variable groups:   𝑦,𝐴   𝜑,𝑦   𝜓,𝑦   𝑥,𝜓′   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)   𝜓′(𝑦)

Proof of Theorem bnj610

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 ⊢ 𝑦 ∈ V
2		bnj610.3	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓′))
3	1, 2	sbcie 3470	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓′)
4	3	sbcbii 3491	. 2 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓′)
5		sbcco 3458	. 2 ⊢ ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)
6		bnj610.1	. . 3 ⊢ 𝐴 ∈ V
7		bnj610.4	. . 3 ⊢ (𝑦 = 𝐴 → (𝜓′ ↔ 𝜓))
8	6, 7	sbcie 3470	. 2 ⊢ ([𝐴 / 𝑦]𝜓′ ↔ 𝜓)
9	4, 5, 8	3bitr3i 290	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  Vcvv 3200  [wsbc 3435

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-v 3202  df-sbc 3436

This theorem is referenced by:  bnj611  30988  bnj1000  31011