Theorem sbcexfi 38580

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

Hypotheses

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

Assertion

Ref	Expression
sbcexfi	⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbcexfi

Step	Hyp	Ref	Expression
1		sbcexfi.1	. . 3 ⊢ Ⅎ𝑦𝐴
2	1	sbcexf 38578	. 2 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
3		sbcexfi.2	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)
4	3	exbii 1867	. 2 ⊢ (∃𝑦[𝐴 / 𝑥]𝜑 ↔ ∃𝑦𝜓)
5	2, 4	bitri 277	1 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦𝜓)

Syntax hints:   ↔ wb 208  ∃wex 1798  Ⅎwnfc 2908  [wsbc 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-v 3455  df-sbc 3745

This theorem is referenced by: (None)