Theorem sbcalfi 33919

Description: Move universal quantifier in and out of class substitution, with an explicit non-free variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)

Hypotheses

Ref	Expression
sbcalfi.1	⊢ Ⅎ𝑦𝐴
sbcalfi.2	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)

Assertion

Ref	Expression
sbcalfi	⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbcalfi

Step	Hyp	Ref	Expression
1		sbcalfi.1	. . 3 ⊢ Ⅎ𝑦𝐴
2	1	sbcalf 33917	. 2 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
3		sbcalfi.2	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)
4	3	albii 1747	. 2 ⊢ (∀𝑦[𝐴 / 𝑥]𝜑 ↔ ∀𝑦𝜓)
5	2, 4	bitri 264	1 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   ↔ wb 196  ∀wal 1481  Ⅎwnfc 2751  [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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436

This theorem is referenced by: (None)