Theorem bnj133 35358

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj133.1	⊢ (𝜑 ↔ ∃𝑥𝜓)
bnj133.2	⊢ (𝜒 ↔ 𝜓)

Assertion

Ref	Expression
bnj133	⊢ (𝜑 ↔ ∃𝑥𝜒)

Proof of Theorem bnj133

Step	Hyp	Ref	Expression
1		bnj133.1	. 2 ⊢ (𝜑 ↔ ∃𝑥𝜓)
2		bnj133.2	. . 3 ⊢ (𝜒 ↔ 𝜓)
3	2	exbii 1880	. 2 ⊢ (∃𝑥𝜒 ↔ ∃𝑥𝜓)
4	1, 3	bitr4i 281	1 ⊢ (𝜑 ↔ ∃𝑥𝜒)

This proof depends on syntax axioms:   ↔ wb 209  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  bnj150  35506  bnj983  35581  bnj984  35582  bnj985v  35583  bnj985  35584  bnj1090  35609  bnj1514  35693